This dissertation reports some first steps towards a compositional account of active inference and the Bayesian brain. Specifically, we use the tools of contemporary applied category theory to supply functorial semantics for approximate inference. To do so, we define on the `syntactic' side the new notion of Bayesian lens and show that Bayesian updating composes according to the compositional lens pattern. Using Bayesian lenses, and inspired by compositional game theory, we define fibrations of statistical games and classify various problems of statistical inference as corresponding sections: the chain rule of the relative entropy is formalized as a strict section, while maximum likelihood estimation and the free energy give lax sections. In the process, we introduce a new notion of `copy-composition'. On the `semantic' side, we present a new formalization of general open dynamical systems (particularly: deterministic, stochastic, and random; and discrete- and continuous-time) as certain coalgebras of polynomial functors, which we show collect into monoidal opindexed categories (or, alternatively, into algebras for multicategories of generalized polynomial functors). We use these opindexed categories to define monoidal bicategories of cilia: dynamical systems which control lenses, and which supply the target for our functorial semantics. Accordingly, we construct functors which explain the bidirectional compositional structure of predictive coding neural circuits under the free energy principle, thereby giving a formal mathematical underpinning to the bidirectionality observed in the cortex. Along the way, we explain how to compose rate-coded neural circuits using an algebra for a multicategory of linear circuit diagrams, showing subsequently that this is subsumed by lenses and polynomial functors.

The theory of belief functions [162, 67] is a modelling language for representing and combining elementary items of evidence, which do not necessarily come in the form of sharp statements, with the goal of maintaining a mathematical representation of an agent's beliefs about those aspects of the world which the agent is unable to predict with reasonable certainty. While arguably a more appropriate mathematical description of uncertainty than classical probability theory, for the reasons we have thoroughly explored in [50], the theory of evidence is relatively simple to understand and implement, and does not require one to abandon the notion of an event, as is the case, for instance, for Walley's imprecise probability theory [193]. It is grounded in the beautiful mathematics of random sets, and exhibits strong relationships with many other theories of uncertainty. As mathematical objects, belief functions have fascinating properties in terms of their geometry, algebra [207] and combinatorics. Despite initial concerns about the computational complexity of a naive implementation of the theory of evidence, evidential reasoning can actually be implemented on large sample spaces [156] and in situations involving the combination of numerous pieces of evidence [74]. Elementary items of evidence often induce simple belief functions, which can be combined very efficiently with complexity O(n + 1).

Noise is inevitably common in digital images, leading to visual image deterioration. Therefore, a suitable filtering method is required to lessen the noise while preserving the image features (edges, corners, etc.). This paper presents the efficient type-2 fuzzy weighted mean filter with an adaptive threshold to remove the SAP noise. The present filter has two primary steps: The first stage categorizes images as lightly, medium, and heavily corrupted based on an adaptive threshold by comparing the M-ALD of processed pixels with the upper and lower MF of the type-2 fuzzy identifier. The second stage eliminates corrupted pixels by computing the appropriate weight using GMF with the mean and variance of the uncorrupted pixels in the filter window. Simulation results vividly show that the obtained denoised images preserve image features, i.e., edges, corners, and other sharp structures, compared with different filtering methods.

We consider a combinatorial Gaussian process semi-bandit problem with time-varying arm availability. Each round, an agent is provided a set of available base arms and must select a subset of them to maximize the long-term cumulative reward. Assuming the expected rewards are sampled from a Gaussian process (GP) over the arm space, the agent can efficiently learn. We study the Bayesian setting and provide novel Bayesian regret bounds for three GP-based algorithms: GP-UCB, Bayes-GP-UCB and GP-TS. Our bounds extend previous results for GP-UCB and GP-TS to a combinatorial setting with varying arm availability and to the best of our knowledge, we provide the first Bayesian regret bound for Bayes-GP-UCB. Time-varying arm availability encompasses other widely considered bandit problems such as contextual bandits. We formulate the online energy-efficient navigation problem as a combinatorial and contextual bandit and provide a comprehensive experimental study on synthetic and real-world road networks with detailed simulations. The contextual GP model obtains lower regret and is less dependent on the informativeness of the prior compared to the non-contextual Bayesian inference model. In addition, Thompson sampling obtains lower regret than Bayes-UCB for both the contextual and non-contextual model.

For robotic decision-making under uncertainty, the balance between exploitation and exploration of available options must be carefully taken into account. In this study, we introduce a new variant of contextual multi-armed bandits called observation-augmented CMABs (OA-CMABs) wherein a decision-making agent can utilize extra outcome observations from an external information source. CMABs model the expected option outcomes as a function of context features and hidden parameters, which are inferred from previous option outcomes. In OA-CMABs, external observations are also a function of context features and thus provide additional evidence about the hidden parameters. Yet, if an external information source is error-prone, the resulting posterior updates can harm decision-making performance unless the presence of errors is considered. To this end, we propose a robust Bayesian inference process for OA-CMABs that is based on the concept of probabilistic data validation. Our approach handles complex mixture model parameter priors and hybrid observation likelihoods for semantic data sources, allowing us to develop validation algorithms based on recently develop probabilistic semantic data association techniques. Furthermore, to more effectively cope with the combined sources of uncertainty in OA-CMABs, we derive a new active inference algorithm for option selection based on expected free energy minimization. This generalizes previous work on active inference for bandit-based robotic decision-making by accounting for faulty observations and non-Gaussian inference. Our approaches are demonstrated on a simulated asynchronous search site selection problem for space exploration. The results show that even if incorrect observations are provided by external information sources, efficient decision-making and robust parameter inference are still achieved in a wide variety of experimental conditions.

Decision trees remain one of the most popular machine learning models today, largely due to their out-of-the-box performance and interpretability. In this work, we present a Bayesian approach to decision tree induction via maximum a posteriori inference of a posterior distribution over trees. We first demonstrate a connection between maximum a posteriori inference of decision trees and AND/OR search. Using this connection, we propose an AND/OR search algorithm, dubbed MAPTree, which is able to recover the maximum a posteriori tree. Lastly, we demonstrate the empirical performance of the maximum a posteriori tree both on synthetic data and in real world settings. On 16 real world datasets, MAPTree either outperforms baselines or demonstrates comparable performance but with much smaller trees. On a synthetic dataset, MAPTree also demonstrates greater robustness to noise and better generalization than existing approaches. Finally, MAPTree recovers the maxiumum a posteriori tree faster than existing sampling approaches and, in contrast with those algorithms, is able to provide a certificate of optimality. The code for our experiments is available at https://github.com/ThrunGroup/maptree.

One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly available and can be approximated conventionally by finite differences. However, the discrete approximations of time derivatives may result in a poor estimation when state data are scarce and/or corrupted by noise, thus compromising the predictiveness of the learned dynamical models. To overcome this technical hurdle, we propose a new method that learns nonlinear dynamics through a Bayesian inference of characterizing model parameters. This method leverages a Gaussian process representation of states, and constructs a likelihood function using the correlation between state data and their derivatives, yet prevents explicit evaluations of time derivatives. Through a Bayesian scheme, a probabilistic estimate of the model parameters is given by the posterior distribution, and thus a quantification is facilitated for uncertainties from noisy state data and the learning process. Specifically, we will discuss the applicability of the proposed method to two typical scenarios for dynamical systems: parameter identification and estimation with an affine structure of the system, and nonlinear parametric approximation without prior knowledge.

Traditional social learning frameworks consider environments with a homogeneous state, where each agent receives observations conditioned on that true state of nature. In this work, we relax this assumption and study the distributed hypothesis testing problem in a heterogeneous environment, where each agent can receive observations conditioned on their own personalized state of nature (or truth). This situation arises in many scenarios, such as when sensors are spatially distributed, or when individuals in a social network have differing views or opinions. In these heterogeneous contexts, the graph topology admits a block structure. We study social learning under personalized (or multitask) models and examine their convergence behavior.

Federated learning (FL) is an emerging approach for training machine learning models collaboratively while preserving data privacy. The need for privacy protection makes it difficult for FL models to achieve global transparency and explainability. To address this limitation, we incorporate logic-based explanations into FL by proposing the Logical Reasoning-based eXplainable Federated Learning (LR-XFL) approach. Under LR-XFL, FL clients create local logic rules based on their local data and send them, along with model updates, to the FL server. The FL server connects the local logic rules through a proper logical connector that is derived based on properties of client data, without requiring access to the raw data. In addition, the server also aggregates the local model updates with weight values determined by the quality of the clients' local data as reflected by their uploaded logic rules. The results show that LR-XFL outperforms the most relevant baseline by 1.19%, 5.81% and 5.41% in terms of classification accuracy, rule accuracy and rule fidelity, respectively. The explicit rule evaluation and expression under LR-XFL enable human experts to validate and correct the rules on the server side, hence improving the global FL model's robustness to errors. It has the potential to enhance the transparency of FL models for areas like healthcare and finance where both data privacy and explainability are important.

The growing use of voice user interfaces has led to a surge in the collection and storage of speech data. While data collection allows for the development of efficient tools powering most speech services, it also poses serious privacy issues for users as centralized storage makes private personal speech data vulnerable to cyber threats. With the increasing use of voice-based digital assistants like Amazon's Alexa, Google's Home, and Apple's Siri, and with the increasing ease with which personal speech data can be collected, the risk of malicious use of voice-cloning and speaker/gender/pathological/etc. recognition has increased. This thesis proposes solutions for anonymizing speech and evaluating the degree of the anonymization. In this work, anonymization refers to making personal speech data unlinkable to an identity while maintaining the usefulness (utility) of the speech signal (e.g., access to linguistic content). We start by identifying several challenges that evaluation protocols need to consider to evaluate the degree of privacy protection properly. We clarify how anonymization systems must be configured for evaluation purposes and highlight that many practical deployment configurations do not permit privacy evaluation. Furthermore, we study and examine the most common voice conversion-based anonymization system and identify its weak points before suggesting new methods to overcome some limitations. We isolate all components of the anonymization system to evaluate the degree of speaker PPI associated with each of them. Then, we propose several transformation methods for each component to reduce as much as possible speaker PPI while maintaining utility. We promote anonymization algorithms based on quantization-based transformation as an alternative to the most-used and well-known noise-based approach. Finally, we endeavor a new attack method to invert anonymization.